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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 797398, 16 pages
Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives
1School of Information and Engineering, Wenzhou Medical College, Zhejiang, Wenzhou 325035, China
2School of Mathematical and Informational Sciences, Yantai University, Shandong, Yantai 264005, China
3Information Engineering Department, Anhui Xinhua University, Anhui, Hefei 230031, China
Received 24 April 2012; Accepted 29 May 2012
Academic Editor: Yonghong Wu
Copyright © 2012 Tunhua Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the singular fractional-order boundary-value problem with a sign-changing nonlinear term , , where , and with and , satisfying , is the standard Riemann-Liouville derivative, is a sign-changing continuous function and may be unbounded from below with respect to , and is continuous. Some new results on the existence of nontrivial solutions for the above problem are obtained by computing the topological degree of a completely continuous field.
Fractional differential equations arise in many engineering and scientific disciplines, and particularly in the mathematical modeling of systems and processes in physics, chemistry, aerodynamics, electrodynamics of complex medium, and polymer rheology [1–6]. Fractional-order models have proved to be more accurate than integer-order models, that is, there are more degrees of freedom in the fractional-order models. Hence fractional differential equations have attracted great research interest in recent years, and for more details we refer the reader to [7–16] and the references cited therein.
In this paper, we consider the existence of nontrivial solutions for the following singular fractional-order boundary-value problem with a sign-changing nonlinear term and fractional derivatives: where and with and , satisfying , is the standard Riemann-Liouville derivative, is a sign-changing continuous function and may be unbounded from below with respect to , and is continuous.
In this paper, we assume that , which implies that the problem (1.1) is changing sign (or semipositone particularly). Differential equations with changing-sign arguments are found to be important mathematical tools for the better understanding of several real-world problems in physics, chemistry, mechanics, engineering, and economics [17–19]. In general, the cone theory is difficult to handle this type of problems since the operator generated by is not a cone mapping. So to find a new method to solve changing-sign problems is an interesting, important, and difficult work. An effective approach to this problem was recently suggested by Sun  based on the topological degree of a completely continuous field. Then, Han and Wu [21, 22] obtained a new Leray-Schauder degree theorem by improving the results of Sun . In , Han et al. also investigated a kind of singular two-point boundary-value problems with sign-changing nonlinear terms by applying the new Leray-Schauder degree theorem obtained in .
To our knowledge, very few results have been established when is changing sign [20–24]. In [20, 21, 23], permits sign changing but required to be bounded from below. In [22, Theorem 1.1], may be a sign-changing and unbounded function, but the Green function must be symmetric and is controlled by a special function , where and . Recently, by improving and generalizing the main results of Sun  and Han et al. [21, 22], Liu et al.  established a generalized Leray-Schauder degree theorem of a completely continuous field for solving -point boundary-value problems for singular second-order differential equations.
Motivated by [20–24], we established some new results on the existence of nontrivial solutions for the problem (1.1) by computing the topological degree of a completely continuous field. The conditions used in the present paper are weaker than the conditions given in previous works [20–24], and particularly we drop the assumption of even function in . The new features of this paper mainly include the following aspects. Firstly, the nonlinear term in the BVP (1.1) is allowed to be sign changing and unbounded from below with respect to . Secondly, the nonlinear term involves fractional derivatives of unknown functions. Thirdly, the boundary conditions involve fractional derivatives of unknown functions which is a more general case, and include the two-point, three-point, multipoint, and some nonlocal problems as special cases of (1.1).
2. Preliminaries and Lemmas
In this section, we give some preliminaries and lemmas.
Definition 2.1. Let be a real Banach space. A nonempty closed convex set is called a cone of if it satisfies the following two conditions:
(1) implies ;
(2) implies .
Definition 2.2. An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.
Let be a real Banach space, the dual space of , a total cone in , that is, and the dual cone of .
Lemma 2.3 (Deimling ). Let be a continuous linear operator, a total cone, and . If there exist and a positive constant such that , then the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue .
Lemma 2.4 (see ). Let be a cone of the real Banach space , and a bounded open subsets of . Suppose that is a completely continuous operator. If there exists such that , then the fixed-point index .
Let be a completely continuous linear positive operator with the spectral radius . On account of Lemma 2.3, there exist and such that where is the dual operator of . Choose a number and let then is a cone in .
Lemma 2.5 (see ). Suppose that the following conditions are satisfied. (A1) is a continuous operator satisfying (A2) is a bounded continuous operator and there exists such that , for all ; (A3) and there exist and such that Let , then there exists such that where is the open ball of radius in .
Remark 2.6. If the operator which satisfies the conditions of Lemma 2.5 is a null operator, then Lemma 2.5 turns into Theorem 1 in . On the other hand, if the operator in Lemma 2.5 is such that there exist constants and satisfying for all , then Lemma 2.5 turns into Theorem 2.1 in  or Theorem 1 in . So Lemma 2.5 is an improvement of the results of paper [20–22].
Now we present the necessary definitions from fractional calculus theory. These definitions can be found in some recent literatures, for example, [26, 27].
Definition 2.8 (see [26, 27]). The Riemann-Liouville fractional derivative of order of a function is given by where , and denotes the integer part of the number , provided that the right-hand side is pointwisely defined on .
Remark 2.9. If with order , then
Lemma 2.10 (see ). (1) If , then
(2) If , then
Lemma 2.11 (see ). Assume that with a fractional derivative of order . Then where , is the smallest integer greater than or equal to .
Noticing that , let by , for , one has
Lemma 2.12. If and , then the boundary-value problem has the unique solution where is the Green function of the boundary-value problem (2.14).
Proof. By applying Lemma 2.11, we may reduce (2.14) to an equivalent integral equation: Note that and (2.17), we have . Consequently the general solution of (2.14) is By (2.18) and Lemma 2.10, we have So, and for , By , combining with (2.20) and (2.21), we obtain So, the unique solution of problem (2.14) is The proof is completed.
Lemma 2.13. The function has the following properties:
(2) , where
Proof. It is obvious that (1) holds. In the following, we will prove (2). In fact, by (2.13), we have This completes the proof.
Now let us consider the following modified problems of the BVP (1.1):
Proof. Substituting into (1.1), by the definition of the Riemann-Liouville fractional derivative and Lemmas 2.10 and 2.11, we obtain that
Also, we have , and it follows from that . Hence, by , (1.1) is transformed into (2.14).
Now, let be a solution of problem (2.26). Then, by Lemma 2.10, (2.26), and (2.27), one has Noticing that which implies that , from (2.27), for , we have
Moreover, it follows from the monotonicity and property of that
Consequently, is a positive solution of problem (1.1).
In the following let us list some assumptions to be used in the rest of this paper.
(H1) is continuous, on any subinterval of (0,1), and
(H2) is continuous.
In order to use Lemma 2.5, let be our real Banach space with the norm and , then is a total cone in .
Define two linear operators by and define a nonlinear operator by
Lemma 2.15. Assume (H1) holds. Then(i) are completely continuous positive linear operators with the first eigenvalue and , respectively.(ii) satisfies .
Proof. (i) By using the similar method of paper , it is easy to know that are completely continuous positive linear operators. In the following, by using the Krein-Rutmann’s theorem, we prove that have the first eigenvalue and , respectively.
In fact, it is obvious that there is such that . Thus there exists such that and for all . Choose such that and for all . Then for ,
So there exists such that for . It follows from Lemma 2.3 that the spectral radius . Thus corresponding to , the first eigenvalue of , and has a positive eigenvector such that
In the same way, has a positive first eigenvalue and a positive eigenvector corresponding to the first eigenvalue , which satisfy
(ii) Notice that for , by and (2.12)–(2.16), we have . This implies that and (see ). Define a function on by
Then is continuous on and for all . So, there exist such that for all . Thus for all .
Let be the dual operator of , we will show that there exists such that In fact, let Then by (H1) and (2.40), we have which implies that is well defined. We state that of (2.42) satisfies (2.41). In fact, by (2.40), (2.41), and interchanging the order of integration, for any , we have So (2.41) holds.
In the following we prove that . In fact, by (2.41) and , we have Take in (2.2). For any , by (2.44), (2.45), we have Hence, , that is, . The proof is completed.
3. Main Result
Theorem 3.1. Assume that (H1)(H2) hold, and the following conditions are satisfied.
(H3) There exist nonnegative continuous functions and a nondecreasing continuous function satisfying
(H4) also satisfies uniformly on , where is the first eigenvalue of the operator defined by (2.34).
Then the singular fractional-order boundary-value problem (1.1) has at least one nontrivial solution.
Proof. According to Lemma 2.15, satisfies . Let
where . It follows from (H3) that is a continuous operator. Note that
Thus from the monotone assumption of on , we have
which implies that
Hence satisfies condition (A1) in Lemma 2.5.
Next take , and for . Then it follows from (H3) that that yields namely, condition (A2) in Lemma 2.5 holds.
From (H4), there exists and a sufficiently large such that, for any , Combining (H3) with (3.12), there exists such that that is, As is a positive linear operator, it follows from (3.14) that So condition (A3) in Lemma 2.5 holds. According to Lemma 2.5, there exists a sufficiently large number such that
On the other hand, it follows from (H4) that there exist and , for any , such that Thus for any with , we have By (3.17), for any , we have Thus if there exist and such that , then by (3.19), we have Therefore, .
But for all by the maximum principle, and attains zero on isolated points by the Sturm theorem. Hence, from (2.42), This is a contradiction. Thus It follows from the homotopy invariance of the Leray-Shauder degree that By (3.16), (3.23), and the additivity of Leray-Shauder degree, we obtain As a result, has at least one fixed point on , namely, the BVP (1.1) has at least one nontrivial solution .
Corollary 3.2. Assume that (H1), (H2), and (H4) hold. If the following condition is satisfied.
(H*3) There exists a nonnegative continuous functions such that then the singular higher multipoint boundary-value problems (1.1) have at least one nontrivial solution.
Corollary 3.3. Assume that (H1), (H2), and (H4) hold. If the following condition is satisfied.
(H3**) There exist three constants , and such that then the singular higher multipoint boundary-value problems (1.1) have at least one nontrivial solution.
Remark 3.4. Noticing that the Green function of the BVP (1.1) is not symmetrical, which implies that the existence of nontrivial solutions of the BVP (1.1) cannot be obtained by Theorem 2.1 in  and Theorem 1 in . It is interesting that we construct a new linear operator instead of in paper  and use its first eigenvalue and its corresponding eigenfunction to seek a linear continuous functional of . As a result, we overcome the difficulty caused by the nonsymmetry of the Green function. In , the nonlinearity does not contain derivatives and a stronger condition is required, that is, must be an even function; here we omit this stronger assumption.
This project is supported financially by Scientific Research Project of Zhejiang Education Department (no. Y201016244), also by Scientific Research Project of Wenzhou (no. G20110004), Natural Science Foundation of Zhejiang Province (No. 2012C31025) and the National Natural Science Foundation of China (11071141, 11126231, 21207103), and the Natural Science Foundation of Shandong Province of China (ZR2010AM017).
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